Moments of the Wigner distribution of rotationally symmetric partially coherent light

The Wigner distribution of rotationally symmetric partially coherent light is considered, and the constraints for its moments are derived. Although all odd-order moments vanish, these constraints lead to a drastic reduction in the number of parameters that we need to describe all even-order moments: whereas in general we have (N + 1)(N + 2)(N + 3)/6 different moments of order N, this number reduces to (1 + N/2)2 in the case of rotational symmetry. A way to measure the moments as intensity moments in the output planes of (generally anamorphic) fractional Fourier-transform systems is presented.     

Reduced forms of the Wigner distribution function for the numerical analysis of rotationally symmetric synchrotron radiation

In an effort to provide a computationally convenient approach to the characterization of partially coherent synchrotron radiation in phase space, a thorough discussion of the minimum dimensionality of the Wigner distribution function for rotationally symmetric sources of arbitrary degrees of coherence is presented. It is found that perfectly coherent, perfectly incoherent and partially coherent sources may all be characterized by a three‐dimensional reduced Wigner distribution function, and some special cases are discussed in which a two‐dimensional reduced Wigner distribution function suffices. An application of the dimension‐reducing formalism to the case of partially coherent radiation from a planar undulator and a circularly symmetric electron beam as can be found in linear accelerators is demonstrated. The photon distribution is convolved over a realistic electron bunch, and how the beta function, emittance and energy spread of the bunch affect the total degree of coherence of the radiation is inspected. Finally the cross spectral density is diagonalized and the eigenmodes of the partially coherent radiation are recovered.     

Fourth order correction to a paraxial Gaussian beam in a rotationally symmetric system of non-spherical surfaces

We formulate the fourth order correction to a paraxial Gaussian beam propagating through a non-spherical surface system that is rotationally symmetric. First we represent the Gaussian beam by a complex-source-point spherical wave (CSPSW). Next we examine the evolution of the CSPSW through the non-spherical surface system by applying a Fresnel–Kirchhoff diffraction integral. We find a ray-optical solution to the diffraction integral in terms of both the conic constant and the aspheric coefficient of fourth order. Then we numerically evaluate the quality factor of the fourth order-corrected beam propagating through either an aspheric mirror or a thin lens made up of two non-spherical surfaces. The fourth order formulas derived here can be useful in determining the quality factor of the Gaussian beam degraded by a rotationally symmetric system of non-spherical surfaces.     

Redistributing energy flow and polarization of a focused azimuthally polarized beam with rotationally symmetric sector-shaped obstacles

The redistribution of transversal energy flow and polarization in the focal field are represented by obstructing an azimuthally polarized beam with rotationally symmetric sector-shaped obstacles. Several energy flow rings that can finally transport the absorptive particles into fixed locations are formed in the focal plane. Furthermore, the local polarization state of the focal field is also modified by use of the rotationally symmetric obstacles. This kind of energy flow may have wide applications in optical trapping and manipulation.     

Rate of equilibration of a one-dimensional Wigner crystal

We consider a system of one-dimensional spinless particles interacting via long-range repulsion. In the limit of strong interactions the system is a Wigner crystal, with excitations analogous to phonons in solids. In a harmonic crystal the phonons do not interact, and the system never reaches thermal equilibrium. We account for the anharmonism of the Wigner crystal and find the rate at which it approaches equilibrium. The full equilibration of the system requires umklapp scattering of phonons, resulting in exponential suppression of the equilibration rate at low temperatures.     

Computer program for the simulation of high intensity beam of charged particles in rotationally symmetric geometry

The computer program for the calculation of particle trajectories of a beam with high density of space charge, in complicated static electric and magnetic fields with rotational symmetry is described. Careful selection of numerical methods and comparatively general arrangement make the program applicable to a wide field of problems. As an example, the accel-decel extraction system of an ion source is solved.     

Some dynamical properties of a two-dimensional Wigner crystal

Results for the static part of the ground state energy of the square and hexagonal two-dimensional Wigner lattices are given. The hexagonal lattice has the lower energy. Phonon dispersion curves and the vibrational zeropoint energy are calculated for the hexagonal lattice. The dielectric susceptibility tensor of a two-dimensional Wigner crystal χ_αβ(q) has been determined in the long wavelength limit in the presence of a static magnetic field perpendicular to the crystal, and explicit expressions have been obtained for the hexagonal lattice. Applying the analysis developed by Chiu and Quinn, the results for the susceptibility have been used to obtain the dispersion relation for the plasma oscillations in the electron crystal on the assumption that the crystal is embedded in a dielectric medium. The dispersion curves have been calculated for differing magnitudes of the applied magnetic field.     

Transition from a two-dimensional superfluid to a one-dimensional Mott insulator

A two-dimensional system of atoms in an anisotropic optical lattice is studied theoretically. If the system is finite in one direction, it is shown to exhibit a transition between a two-dimensional superfluid and a one-dimensional Mott insulating chain of superfluid tubes. Monte Carlo simulations are consistent with the expectation that the phase transition is of Kosterlitz-Thouless type. The effect of the transition on experimental time-of-flight images is discussed.     

Beam-propagation method based on the Wigner transform: a new formulation

A new formulation of the classical scalar beam-propagation method is derived by use of the Wigner transform. The new method is faster than the classical beam-propagation method because no Fourier transform must be computed. An example given to illustrate the proposed method shows additional advantages.     

Structural instabilities of a two-dimensional Wigner crystal in a lateral superlattice

We consider the instability of a two-dimensional Wigner crystal in a short-period lateral superlattice. To find instabilities, we calculate the phonon spectrum of the electron lattice deformed by a periodic potential. We show that one of the transverse modes of the deformed electron crystal becomes soft when the electron density exceeds a critical value. This can result in a phase transition with the formation of a charge-density wave.     

Wigner function of a two-dimensional electron gas in a magnetic field

The Wigner function of a two-dimensional electron gas in an arbitrary magnetic field perpendicular to the plane in which the electrons are confined is constructed rigorously. The function is useful in taking various statistical averages and illuminates the roles played by the hyperbolic functions of the field which appear in the expressions of the susceptibility and other physical quantities.     

Spatial characterization of the transversal structure of rotationally symmetric pulsed beams

A simple analytical model is proposed to describe the transversal spatial structure of a tridimensional rotationally symmetric pulsed beam. The spatial behaviour of the pulse amplitude is shown to be linked to its (measurable) second- and higher-order intensity moments, namely, beam width, quality parameter and kurtosis. As an illustrative experimental example, this model has been applied to high-quality TEA CO2 laser pulses. This revised version was published online in November 2006 with corrections to the Cover Date.     

Wigner crystallization of electrons in deep traps in a two-dimensional dielectric

A two-dimensional model is used to examine the spatial distribution of electrons in deep traps in a two-dimensional dielectric. When the trap concentration is much higher than the trapped electron concentration, Coulomb repulsion leads to the formation of a two-dimensional quasi-periodic hexagonal lattice of localized electrons (Wigner glass).